How can I scale object to fit another object that shares the same shape but are diffrent size.
I got a solution for you, but, depending on object type, it might take longer times to complete as it's not automated.

You take one of the vertices of the outer ring(the inner ones that you want to match up). Than you hold down SHIFT key and select it's corresponding vertex where you want it to be placed(on the inner ring). After than press Alt + M and choose "At Last" option. This way the vertex will be automatically placed at it's desired position and the duplicate vertices will be removed too. Now you can repeat the same process for the rest of the vertex pairs.

I hope this helps! :)